Universal Structural Characteristics of Planar Granular Packs

Abstract

Dependence of structural self-organization of granular materials on preparation and grain parameters is key to predictive modelling. We study 60 different mechanically equilibrated polydisperse disc packs, generated numerically by two protocols. We show that, for same-variance disc size distributions (DSDs): 1. the mean coordination number of rattler-free packs vs. the packing fraction is a function independent of initial conditions, friction and the DSD: 2. all quadron volume and cell order distributions collapse to universal forms, also independent of the above. We conclude that, contrary to common wisdom, equilibrated granular structures are determined mainly by the packing protocol and higher moments of the DSD.Self-organization of granular systems into solids is of primary importance in science and engineering [1][2][3][4][5][6]. A central quest in the field is for relations between the physical properties of mechanically stable many-grain packs and their redemergent structural characteristics. Yet, there is so far no systematic way quantify such relations. Moreover, different structural properties often correlate well with different physical properties. For example, void size distribution and connectivity correlate well with permeability to flow, which is relevant to underground water, pollutant dispersion and oil extraction [2]; catalysis and heat exchange between the grains and a fluid in the void space are more sensitive to the solid-void surface distribution [3]; and the mechanics of granular solids is governed by an interplay between the structure and the force transmission through the intergranular contacts [4]. The problem is complicated by the many observations that the structural characteristics depend on the distributions of grain shapes, sizes, intergranular friction coefficients and preparation history. A key to progress is the identification of emergent structural properties that are independent of some of these parameters. The aim of this paper is to identify such universality and show that, contrary to current belief, the distributions of a number of structural features of random packs of planar (2D) granular solids collapse to common forms for all intergranular friction coefficients, a wide range of initial preparation states, at least two preparation protocols, and several same-variance grain size distributions. We analyse the structure of numerically generated and mechanically equilibrated 2D polydisperse disc packs, using the quadron-based structural description [4,7,8]. In this method, illustrated in Fig. 1, one first defines the centroids of grains (discs) g and cells c as the mean position vectors of the contact points around them, respectively. Next, the contact points around every disc are connected to make polygons, whose edges are vectors, r gc , that circulate the grain in the clockwise direction. Then, one extends vectors R gc from the disc centroids g to the cell centroids c that surround them. A quadron is the quadrilateral whose diagonals are the vectors r gc and R gc . In th...

Fundamental structural characteristics of planar granular assemblies: Self-organization and scaling away friction and initial state

“…We argued that the structural features studied here should be very little affected, if at all, by the stress microstates and the angoricity [24,25,29] and our results appear to support this argument. Unavoidably, the different systems and samples we studied could not be generated under exactly the same distribution of external forces and each configuration of stress microstate could, in principle, affect the partition function (1). Yet, the collapses we observed appeared to be sample-independent, suggesting insensitivity of the self-organised structural characteristics to the angoricity.…”

“…We quantify in detail this dependence and the deviation it effects from exact collapse for these cells.(vi) We argue that our results support strongly the view that ensemble granular statistical mechanics does not satisfy the uniform measure assumption of conventional statistical mechanics. Results (i)-(iv) have been reported in [1] and are reviewed and elaborated on here. The results in (v) and the argument (vi) are new.…”

“…Regarding the grain centres as vertices and the lines connecting centres of touching grains as edges, we have N V = N , N E = Nz/2, which gives N C = N (z − 2) /2. Noting then that N Cē = 2N E , we have [1,12]…”

“…The experiments consist of preparing and compacting isotropically two-dimensional granular assemblies of polydisperse frictional discs and analysis of the emergent statistical properties of quadrons -the basic structural elements of granular solids. The focus on quadrons is because the statistics of their volumes have been found to display intriguing universal-like features [1]. The dependence of the structures and of the packing fraction on the inter-granuar friction and initial state are analysed and a number of significant results are found.…”

Scaling Theory for the Frictionless Unjamming Transition

“…1). This construction [27] bears similarities to the "quadron" framework [32][33][34].We then assign the polygonal grain areas to triangular units α c (normalized by the size of each disk), uniquely associated with each contact c. This defines a reliable local order parameter for the unjamming transition [27]. The probability distribution of these areas displays divergences at well defined values of α that become sharper as the transition is approached.…”